Advances in Applied Probability

Percolation and connectivity in AB random geometric graphs

Srikanth K. Iyer and D. Yogeshwaran

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Given two independent Poisson point processes Φ(1), Φ(2) in Rd, the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

Article information

Adv. in Appl. Probab., Volume 44, Number 1 (2012), 21-41.

First available in Project Euclid: 8 March 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C80: Random graphs [See also 60B20]
Secondary: 82B43: Percolation [See also 60K35] 05C40: Connectivity

Random geometric graph percolation connectivity wireless network secure communication


Iyer, Srikanth K.; Yogeshwaran, D. Percolation and connectivity in AB random geometric graphs. Adv. in Appl. Probab. 44 (2012), no. 1, 21--41. doi:10.1239/aap/1331216643.

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